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COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM*

Published online by Cambridge University Press:  13 May 2014

JOHN T. BALDWIN
JOHN T. BALDWIN*
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE M/C 249, UNIVERSITY OF ILLINOIS AT CHICAGO, CHICAGO, IL 6067, USA
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Abstract

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We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.

Keywords

Categoricityformalizationmodel theoryfoundationalismcompleteness

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Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 1 , March 2014, pp. 39 - 79
Copyright
Copyright © The Association for Symbolic Logic 2014 

Footnotes

*

I realized while writing that the title was a subconscious homage to the splendid historical work on Completeness and Categoricity by Awodey and Reck [6].

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